Geoscience Reference
In-Depth Information
Figure 11.5
Typical shape of
large rain droplets when they
have achieved terminal
velocity. (From Smith, 1993,
after Doviak and Zrnic, 1984,
published with permission.)
8 mm
7.35 mm
5.80 mm
5.30 mm
3.45 mm
2.7 mm
Rainfall rates and kinetic energy
The terminal velocity of a perfectly spherical water droplet is proportional to the
square root of its diameter. However, as raindrops fall they vibrate and, being fluid,
they deform and flatten (Fig. 11.5) and, as a result, for drops with average dimen-
sions of the order 0.8 to 4 mm, the observed terminal velocities of raindrops are
described by:
vD
D
0.67
(11.4)
(
)
=
3.86
where
v
(
D
) is the terminal velocity in m s
−1
for a droplet with average dimension
of
D
mm.
The rainfall rate,
R
in mm s
−1
, is given by the integral of the product of raindrop
volume with the terminal velocity of the drop, weighted by the number of drops of
given diameter per unit volume, i.e., by:
∞
⎡
⎤
⎛
3
⎞
π
D
()()
∫
R
=
N D v D
dD
(11.5)
⎢
⎜
⎟
⎥
6
⎢
⎝
⎠
⎥
⎣
⎦
0
in which
N(D
) and
v
(
D
) are given by equations (11.3) and (11.4), respectively.
In a similar way, the kinetic energy flux of the rain falling to the ground,
E
KE
, is
proportional to half the integral of the product of raindrop mass with terminal
velocity squared, again weighted by the number of drops of given diameter per
unit volume, i.e., by:
∞
⎡
⎛
3
⎞
⎤
π
D
() ()
∫
E
N D v D dD
2
(11.6)
=
r
⎢
⎥
⎜
⎟
KE
w
12
⎝
⎠
⎣
⎦
0
The equations used in hydrology to estimate soil erosion due to the impact of
raindrops are an empirical form of the relationship between equations (11.5) and
(11.6) with an assumed drop size distribution.
Forms of frozen precipitation
Numerous forms of frozen precipitation are recognized (Table 11.2), but as is the
case for liquid precipitation, two broad classes of frozen precipitation can be rec-
ognized. These are associated with the mechanisms by means of which the frozen
