EAS 6140 Thermodynamics of Atmospheres and Oceans
Precipitation Microphysics
1. Terminal velocity is determined from a balance of which two forces?
gravity and friction
2. Match the droplet size range to the terminal velocity functional dependence on r:
__c____ r1/2
__a____ r
__b____ r 2
a) 40 m – 0.6 mm
b) < 40 m
c) 0.6 mm – 2 mm
3. Identify approximate terminal velocities for the following drop diameters (Fig 8.2):
10 m ____0.2 m/s______.
100 m ___1 m/s_______.
1000 m __6 m/s________.
3 mm ____9 m/s________________.
4. Consider Fig 8.3. Why do you think that graupel has the highest values of terminal
velocity? Graupel is more solid, spherical and dense than snow crystals which have large
surface areas with less density creating drag as they grow
6. Consider Fig. 8.3. Why does the terminal velocity for graupel increase with
increasing size, while terminal velocity for dendritic crystals is essentially constant with
size? As graupel grows it stays relatively small and dense allowing for acceleration vs
dendritic crystals which grow in a less dense increasing area form allowing it increases
the drag
7. Collision efficiencies between two cloud drops will be largest for the following
combination of sizes (refer to figure 8.5) :
a) 10 m and 20 m
b) 80 m and 20 m
c) 40 m and 10 m
8. Consider Fig. 8.5. How can the collision efficiency exceed 1?
Large particles can collect other large particles through “wake capture”
Below are listed some of the physical processes that contribute to the growth of cloud
particles:
a) diffusional growth of water droplets
b) diffusional growth of ice crystals
c) collision/coalescence between water drops
d) collision/coalescence between ice crystals
e) collision/coalescence of ice crystals with water drops
Many of these choices can be eliminated as a function of the phase and size of the
droplet/crystal in question
In the following situations, which of the above processes will contribute to the most rapid
growth of the condensed particles?
9. Small ice crystals, about 50 microns, coexisting with small water drops. ___E_______
You should automatically eliminate choices that only refer to water or ice. This leaves
only E.
10. A cumulus cloud with a cold top has large (1 mm) ice crystals coexisting with large
(1 mm) raindrops. __E________
11. The cloud is completely glaciated at a temperature of about -30oC, with S < 1. Ice
crystals are columnar with 3 mm long axis. __B____________
A glaciated cloud that is subsaturated @ low temperatures will not grow by diffusion of
water droplets (we can also eliminate c and e since the cloud is sub saturated no water
exists). This leaves B and D. With a 3mm long axis, it is sufficient to say that this is a
relatively large particle. As stated on page 211, larger particles fall faster than small
particles and may collide with a smaller, slower particle that is in its path. However,
because the particles are all columnar they will collide but will not coalesce; they will not
grow through this pathway unless they are dendritic. Answer is B.
12. A spectrum of water drops ranging from 1 to 50 m __C__________
13. Precipitation type matching:
_b____liquid, 400 m diameter
a. graupel
_c____liquid, 3 mm diameter
b. drizzle
_e____ice, diameter 20 mm, density 850 kg m-3
c. rain
_a____ice, diameter 4 mm, density 850 kg m-3
d. snow
_d____ice, diameter 20 mm, density 250 kg m-3
e. hail
14. A hailstone is more likely to melt in the subcloud region if the relative humidity of
the subcloud air is (high, low) Explain`
High humidity prevents evaporative cooling from lowering the temperature of the
subcloud air
15. Write an expression for the mass of a liquid cloud drop of radius r.
4
m   r 3 l
3
16. Write an expression for the liquid water mixing ratio of N drops of radius r.
 4
wl  l  r 3 N
a 3
17. Write an expression for the liquid water mixing ratio for a spectrum of cloud drops, n(r)
 4
wl  l   r 3n(r )dr
a 0 3
18. A cloud drop size distribution has the form
n(r) = No r -(+1)
over the radius range r=0 to R, where No and  are constant. Evaluate the liquid water
mixing ratio for this cloud.
 R4
wl  l   r 3 N 0 r  ( 1) dr
a 0 3
wl 
l
a

R
0
4
 N 0 r 2 dr
3
4 
R 3
wl   l N 0
3 a
3
19. Assuming that the liquid water mixing ratio is constant with height in the cloud,
evaluate the liquid water path for a cloud of depth h.
zt
Wl    a wl dz
zb
h
Wl    a wl dh
0
Wl   a wl h now sub in from 18
4
R 3
Wl  l N 0
h
3
3
20. If all the liquid water in the cloud were to fall out, what would the depth of the
precipitation be?
W
Pad  l now sub in from 19
l
4
R 3
Pad   N 0
h
3
3
21. Calculate the time required for the diameter of a spherical snowflake to increase from 1 mm
to 1 cm if it grows by aggregation as it falls through a cloud of small ice crystals present in an
amount 1 g kg-1. You may assume that the collection efficiency is unity, that the density of the
snowflake is 100 kg m-3, and the difference in the fall speeds of the snowflake and ice crystals is
constant and equal to 1 m s-1.
Problem 8.2 Ans: t=1412 s
Use equation 8.2
dR   R  r 
3
  
 uT R   uT r n(r )r E ( R, r )dr
dt
30  R 
Assuming R>> r, E(R,r) = 1 and [uT(R) – uT(r)] is constant so that dR/dt reduces
R
dR 
  n(r )r 3 dr
dt 3 R 0
R
2