ESCI 340 – Physical Meteorologoy
Cloud Physics Lesson 6 – Precipitation
References: A Short Course in Cloud Physics (3rd ed.), Rogers and Yau
Microphysics of Clouds and Precipitation (2nd ed.), Pruppacher and Klett
Reading: Rogers and Yau, Chapter 10
RAINFALL RATE AND DROP-SIZE DISTRIBUTION
Imagine a volume of air shown below.
ο The mass of droplets having diameters of between D and dD per unit volume
is dM (M has units of kg m−3, and is called the precipitation water content).
ο The time it takes for this mass of droplets to fall past a horizontal plane is
dτ = dz u
where u is the terminal velocity of the droplets of diameter D.
ο The vertical flux of water mass contained in these droplets is the mass
contained in these droplets divided by the area divided by time. Therefore,
the flux of water mass contained in droplets having diameters between D and
dD is
dF =
dM (dx dy d z ) dz
mass
=
=
dM = u dM .
(dx dy ) dτ
area × time
dt
ο If we know the size distribution of the droplets then
dM =
π
6
ρ L D 3 nd ( D )dD
so that we have
dF =
π
6
ρ LuD 3 nd ( D ) dD .
The total mass flux from all the droplets is found by integrating over all
diameters
F=
∞
π
ρ L ∫ u ( D) D3nd ( D) dD .
6
0
F is the mass of liquid water falling through a unit horizontal area per unit time,
so it is a measure of how much precipitation is striking the ground.
To convert this into a depth of water accumulated on the ground we use the
following dimensional analysis:
flux =
density × volume density × depth × area density × depth
mass
=
=
=
area × time
area × time
area × time
time
so that we have the following relationship between rainfall rate (R, with units of
accumulated depth per time) and flux,
F = ρLR .
Thus, the rainfall rate is given by
R=
π
∞
u ( D) D
6∫
3
n( D) dD .
0
ο Using the mean-value theorem of calculus we can write this as
R=
π
6
∞
u ∫ D3 nd ( D) dD
0
where u is the volume-weighted mean terminal velocity for the population as
a whole,
∞
∞
u = ∫ u ( D) D nd ( D) dD
3
0
∫ D n ( D) dD .
3
d
0
THE MARSHALL-PALMER DROP-SIZE DISTRIBUTION
Raindrop size distribution are often given by the Marshall-Palmer distribution,
nd ( D ) = n0 exp(−ΛD ) .
n0 is called the intercept parameter, and Λ is called the slope factor.
ο The intercept factor is a constant, and is often given a value of n0 = 0.08 cm−4.
For the Marshall-Palmer distribution the rainfall rate is
2
R=
π n0 u
Λ4
.
Thus, if we know the slope factor of the size distribution, and the average
terminal velocity, we can find the rainfall rate. Conversely, if we know the
rainfall rate and the average terminal velocity we can find the slope factor via
14
Λ = (π n0 u ) R −1 4 .
Empirically, the slope factor has been found to depend on rainfall rate via
Λ = 41R −0.21
where R has units of mm/hr and Λ has units of cm−1. This is very close to the
theoretically derived equation.
TYPES OF PRECIPITATION
Rain – Drops of water falling from a cloud, and having a diameter of greater
than 0.5 mm
Drizzle – Liquid water drops having a diameter of less than 0.5 mm. You can
often tell the difference between rain and drizzle because drizzle usually doesn’t
cause ripples in standing water puddles.
Snow – Ice crystals or aggregates of ice crystals. The shape of snowflakes varies
with the temperature at which they are formed.
Snow grains – frozen equivalent of drizzle (from stratus clouds). Diameter < 1
mm.
Sleet (also called ice pellets) – Sleet is frozen raindrops. If greater than 5 mm in
diameter, it is called hail (see note under hail).
Snow pellets (also called graupel) – larger than snow grain, but have diameter < 5
mm. Snow pellets are crunchy and break apart when squeezed. Usually fall in
showers from cumulus congestus clouds. If greater than 5 mm they are called
hail (see note under hail).
Hail – Hail begins as a snowflake that partially or completely melts, and then
refreezes. But, instead of immediately falling to the ground, it gets caught in an
updraft and can make several trips up and down through the cloud, each time
accumulating more ice. Hail is only formed in very strong thunderstorms
3
(cumulonimbus clouds). Hail has diameters > 5 mm. If smaller, it is either snow
pellets or ice pellets, depending on its hardness and crunchiness (see note below).
ο U.S. record hailstone fell in Aurora, Nebraska on June 22, 2003. It was at
least 7 inches in diameter and had a circumference of 18.75 inches!
U.S. record hailstone. NOAA Photo used with permission.
ο NOTE: It is sometimes difficult to differentiate between hail, snow pellets,
and sleet. Here are some rules to follow:
If it has a diameter larger than 5 mm it is hail.
If it has a diameter less than 5 mm, and is transparent and solid it is sleet.
If it has a diameter less than 5 mm, is not transparent, and crunches
when squeezed, it is snow pellets.
Glaze – Also called freezing rain, glaze forms when supercooled raindrops strike
an object and instantly freeze on impact.
Rime – Forms in a manner similar to glaze, only it is caused by the freezing of
supercooled cloud droplets rather than supercooled raindrops. It often forms
feathery ice crystals on trees.
4
EXERCISES
1. If a droplet population consists of droplets of a single size (D = 2 mm,
u = 6.49 m/s), and the droplet density is 26 drops per m3, what is the rainfall rate
in mm/hr?
2. Show that for the Marshall-Palmer drop-size distribution the slope factor and the
precipitation water content are related via
14
Λ = (πρ L n0 M ) .
3. Show that for the Marshall-Palmer drop-size distribution the rainfall rate is
R=
π n0 u
Λ4
.
4. In the formula for rainfall rate,
π
∞
u D3 nd ( D) dD ,
6 ∫0
u is the volume weighted mean terminal velocity. If instead you used the mean
terminal velocity, would the calculated rainfall rate be to low, or too high?
Explain.
R=
5