298
P. Joe
Figure 12.9: Terminal velocities for various crystal types.
For hail, Matson and Huggins (1980) found:
Wt = 3.62Do. 5
(12.17)
for D in mm. At 1.5 km, R is about 10% larger than at the surface due to velocity effects.
Derived relationships
Under the assumption of an exponential distribution, relationships between Z, M and R can
be derived.
7rp (00
M = -f 10 D3 N(D)dD = 7rpwNo/ A 4 = 7rpwNoDci/3.67 4
(12.18)
and the median volume diameter is Do = 3.67/ A.
(12.19)
and
7rpw (00 3 ( ) ( )
7r No [
10.3
1 7r NoDci [
10.3
R = -6- 10 D N D Wt D dD = A4 9.65 (1 + 600/A)4 = 3.674 9.65- (1 + 600 x D o /3.67)41
(12.20)
in 81 units.
12.6.3 Z - R Techniques
Empirical Z - R relations and the variations from storm to storm and within individual storms
have been the subject of many studies over the past forty years. The rainfall rate (R), a
product of the mass content (M) and the fall velocity (Wt) in a radar volume. This is roughly