## Fall speed of hydrometeors

 B. Geerts 4/'00

'Hydrometeors' include all types of precipitation (rain,snow, hail, graupel, sleet …). Droplets found in clouds or fog, as well as ice crystals in high clouds (like cirrus), are generally not included in the term, because their fallspeed is negligible. That is because their weight is too small relative to their surface area (1).

When hydrometeors fall, an equilibrium quickly establishes between two forces, gravity (downward) and aerodynamic drag (upward). The resulting velocity is called the terminal velocity, or simply fallspeed, Vt. The gravitational force is proportional to the drop mass m, hence the 3rd power of its diameter D, while the frictional force is proportional to the cross-sectional area A of the drop, hence the 2nd power of D. The force balance is as follows:

m g = Cd r Vt2 A

where g is the gravitational acceleration, r the air density, and Cd the drag coefficient. The problem is that Cd is not constant; rather, it is a function of both diameter D, fallspeed Vt, and the kinematic viscosity of the air. The latter in turn is variable; it depends on the eddy characteristics of the air flowing around the falling drop. Hence viscosity cannot be expressed in a simple formula. The result is that the best determination of the fallspeed is an empirical one. Foote and Dutoit (1969) proposed this relationship for raindrop fallspeed Vt (in m/s) as a function of D (in mm), for 0.1 mm < D < 6 mm (2):

Vt = [-0.193 + 4.96 D - 0.904 D2 + 0.0566 D3] exp(z/20)

The factor exp(z/20), where z is height in km, accounts for the decrease in density (and hence drag) with height in the atmosphere. Drops larger than 3 mm have a good chance of breaking up into smaller drops. The break-up probability rapidly increases at diameters around 5 mm. Some typical fallspeeds for liquid drops are shown in Table 1.

 diameter: mm fallspeed: m/s 0.001 0.0003 0.01 0.03 0.1 0.27 0.2 (cloud) 0.72 0.3 (cloud) 1.2 0.8 (drizzle) 3.3 0.9 (drizzle) 3.7 1.8 (rain) 6.1 2.2 (rain) 6.9 3.2 (drop breaks up) 8.3 5.8 (ditto) 9.2

Table 1: Terminal velocities for drops of various sizes.

Hail can falls much faster, because its diameter can be larger. Its fallspeed is approximately given by 1.4 D0.8 at sea level, the exact relationship depends on hail density and shape. For instance, a large hailstone of 8 cm (D=80 mm) weighs about 0.7 kg and falls at 48 m/s ! Snow flakes of any size falls at about 1 m/s, slightly faster aloft because of the density correction [exp(z/20)]. The fallspeed of snow increases only slightly with snow flake size is because of the intricate shape of snow : the surface area of a growing crystal increases about as rapidly as its weight. Fall speed relations for various types of unrimed snow crystals are as follows (Vt in m/s, D in mm) (3):

Vt = a Db , where 0.7 < a < 1.2 and 0.11 < b < 0.16

Rimed snow and graupel will fall faster than snow that did not accrete any supercooled droplets.

References

1. Beard, K.V., 1976: Terminal velocity and shape of cloud and precipitation drops aloft. J. Atmos. Sci., 33, 851-864.
2. Foote, G.B. and DuToit, P.S., 1969: Terminal velocity of raindrops aloft. J. Appl. Meteor., 8, 249 253.
3. Bohm, J.P., 1989: A general equation for the terminal fall speed of solid hydrometeors. J. Atmos. Sci., 46, 2419-2427.