- 2.2.1 Raindrop size
- 2.2.2 Drop velocity
- 2.2.3 Stopping distance and dispersion
- 2.2.4 Raindrop spectrum and rain intensity

2.2 Rain

Precipitation occurs when cloud particles, which grow in complex processes like condensation and aggregation, reach such a size that their falling velocity is larger than the upward wind speed in the air. Precipitation is called rain when its particles are liquid water at ground level. Apart from its complex formation in clouds, rain is basically a population of falling drops interacting with each other (collision, breakup) and with their environment (wind, evaporation).

General literature on rain can be found in for example [Pruppacher and Klett 1978]. For climatological information on rain one can refer to [Buishand and Velds 1980] for the Netherlands.

In a first approximation, the minimum size of raindrops falling on the ground depends on vertical wind speeds in clouds. In clouds with updraughts of less than 50 cm s, drops of 0.2 mm (terminal velocity of 70 cm s) and more will fall out. In air of 90% humidity such a drop can fall 150 m before total evaporation and thus reach the ground. A drop of 1 mm can fall 40 km. Rain which mainly consists of drops of 0.1 mm diameter, is called drizzle, and is produced by low layer clouds.

The maximum diameter of raindrops is about 7 mm, because larger drops will break apart during the fall.

Only drops of diameters of less than 0.3 mm are nearly perfect spheres at terminal (falling) velocity. Therefore for larger drops one can not unambiguously describe the shape by one length. This problem is solved by the definition of a equivalent diameter: the diameter of a sphere with the same volume as the deformed drop.

Falling drops of (equivalent) diameters of 0.3 to 1 mm resemble oblate spheroids. Drops larger than 1 mm resemble oblate spheroids with flat bases (figure 2.5).

In the following sections and chapters, the term ``diameter'' should be understood as the ``equivalent diameter''. Moreover, one should keep in mind that raindrop diameters range from 0.1 mm to 7 mm. The number of raindrops per drop size in rainfall (``raindrop spectrum'') is discussed in section 2.2.4.

2.2.2 Drop velocity

The velocity of a drop depends on gravitation and drag due to wind speed.
The motion of a drop can be described by:

Figure 2.6 shows the drag coefficient as function of the Reynolds number. The function of [Morsi and Alexander 1972] was obtained from fitting a large amount of laboratory data for different kinds of particles, from different references. Data on falling water drops in still air obtained by [Gunn and Kinzer 1949] have been included in the graph. For falling drops mm (10), the drag coefficient is underestimated by [Morsi and Alexander 1972].

The terminal velocity is the maximum vertical velocity which a drop reaches. It is, in other words, the velocity when the gravitational force equals the drag force. The usual assumption that the vertical velocity approximately equals the terminal velocity, is thus only valid in wind flow with zero vertical wind velocity. Generally this is a good approximation for the undisturbed wind flow far from obstacles; near buildings vertical wind velocity influences the drop velocity. Strictly speaking, the horizontal wind component could also influence the falling velocity because it can deform the shape of a raindrop and thus the drag.

[Gunn and
Kinzer 1949] presented a table with terminal velocity data as
a function of drop diameter, measured in laboratory. We fitted these data with
the following function:

Often one applies such a relation obtained from laboratory experiments in still air at an air pressure of 760 mm Hg, a temperature of 20C and a relative humidity of 50%. In figure 2.7 the relation given by eq. 2.11 is plotted, with terminal velocities calculated directly from the equation of motion (eq. 2.10) and drag coefficients of [Morsi and Alexander 1972]. The latter method overestimates the terminal velocity of bigger drops ( mm), because the used drag coefficient function does not take drop deformation into account.

2.2.3 Stopping distance and dispersion

The stopping distance
is the distance travelled by
a drop as a result of its inertia, after the driving force is suddenly
taken away [Fuchs 1964]. In our case, the driving force is the wind
speed. The stopping distance is a function of the drop diameter and
the initial speed of the drop and can be calculated by use of
equation 2.10 with the conditions , and
. The solution is:

Figure 2.8a shows the stopping distance for various initial speeds (cf. wind speeds) and various drop diameters, calculated with values of given by [Morsi and Alexander 1972]. The same figure gives results calculated with values of [Gunn and Kinzer 1949], as far as the limited range of Reynolds numbers allowed for (cf. figure 2.6). Figure 2.8b shows the same information as figure 2.8a, but now condensed into two lines, as is obvious from eq. 2.12.

The stopping distance is a measure for the ability of raindrops to follow changes in wind speed and direction. In general, in eddies where wind flow is curved and accelerated, the stopping distance can characterise the dispersion of raindrops, i.e. the raindrop trajectory deviations from the wind pattern. Raindrops with a certain will follow eddies with a certain minimum dimension, say , but eddies with a dimension smaller than will hardly or not effect the raindrop trajectory. We will assume here that approximately equals .

The integral length scale, , is an estimate of the dimensions of the largest eddies in a neutral homogeneous boundary layer. Estimates range from 50 to 200 m [Geurts 1997]. Figure 2.8 shows that for wind speeds up to 10 m s the stopping distance is less than , and that the drops will follow such eddies. Close to buildings we will find smaller eddies, for example a frontal vortex with the dimension of the building height (see section 2.1.4 for other typical values).

Without a numerical model it is not easy to estimate how the raindrops
will be dispersed by eddies other than or a frontal vortex.
[]p. 201]crowe:1998 cite an estimation for the length
scale of a turbulent eddy within the framework of the -
turbulence model for the fluid:

(2.14) |

Although the relation is only a crude approximation, we conclude that dispersion of raindrops due to the turbulence of the wind is very likely an important factor.

2.2.4 Raindrop spectrum and rain intensity

Rain is characterised by raindrop spectra. In meteorology raindrop spectra (also called ``raindrop distributions'') are often expressed in a number of drops per (volume) unit of air per (equivalent) drop diameter. It is here called ``raindrop number concentration spectrum'', denoted by and its unit is m m.

For our purposes, it is more practical to express
raindrop spectra in terms of mass fluxes on or through a certain area. First
we define a raindrop mass concentration spectrum [kg m m]:

Subsequently, a raindrop mass flux spectrum
[kg m s m] through the horizontal is defined by:

Relation 2.16 is only valid (1) if the vertical velocity of every drop equals the terminal velocity (and is not affected by vertical wind, turbulence and raindrop interaction), and (2) if the raindrop spectrum does not depend on time and location (i.e. in stationary rainfall). [Uijlenhoet and Stricker 1999] gave an extensive overview of different definitions of raindrop spectra and their integral quantities.

Raindrop mass flux spectra give a better impression of
the contribution of every drop size to the total rain amount. This amount
is expressed by the horizontal rain intensity [mm h]
and is defined by the amount of rain water falling through a horizontal
plane per hour in the undisturbed wind flow, and equals
the summation of the mass fluxes of all drops falling through the plane:

(2.17) |

Three other quantities are sometimes used, namely (1) the liquid water
content [kg m], i.e. the amount of liquid water per unit of air:

(2.18) |

and (3) the median drop size [m]:

Three frequently used empirical formulae for raindrop spectra are presented in the following.

**[Marshall and
Palmer 1948] spectra**
Probably the most widely used empirical description of raindrop spectra
is that of [Marshall and
Palmer 1948]:

[Marshall and
Palmer 1948] obtained and from measurements amongst
others by
[Laws and
Parsons 1943], who describe in detail the measurement and
averaging methods used at that time:

Examples of [Marshall and Palmer 1948] spectra are plotted in figure 2.9a. We converted these raindrop size spectra into raindrop mass flux spectra and the result is plotted in figure 2.9b. Comparison of figures a and b shows that a mass flux spectrum gives a better impression of the contribution of every drop size to the total rain amount on the horizontal.

Subsequent studies pointed out that the parameters and vary widely (and even during rainfall). One has tried to classify these parameters by the type of rain, as e.g. in [Joss and Waldvogel 1969] (table 2.2).

type of rainfall | - relationship | |

[m mm] | [ mm, mm h] | |

drizzle | 30 000 | |

widespread | 7 000 | |

thunderstorm | 1 400 |

[Marshall and Palmer 1948] obtained the parameterisation 2.22 by analysing rainfall with drops of 1 to 3.5 mm diameter and with rain intensities of 1 to 23 mm h. It is therefore often assumed that this parameterisation overestimates of the number of drops below 1 mm diameter. Therefore, other spectra (e.g. gamma distributions) were proposed. An overview is given in [Sempere Torres and Porrà 1994].

**[Ulbrich 1983] spectra**
The raindrop spectrum of [Ulbrich 1983] is defined as:

The Ulbrich spectrum includes an extra parameter (), which should take variations in (of eq. 2.21) into account. E.g. [Waldvogel 1974] showed that large and sudden changes in can occur from moment to moment within a given rainfall type. Figure 2.10 shows examples of Ulbrich spectra. Note that for the Ulbrich spectrum equals the Marshall-Palmer spectrum.

**[Best 1950] spectra**
The third type of raindrop spectrum presented here, was proposed by
[Best 1950]. The spectrum is expressed by:

with = the fraction of liquid water in the air comprised by drops with diameter less than [mm], = the horizontal rain intensity in mm h, and = the amount of liquid water per unit of air in mm m.

The values of the constants , , , and were obtained from results of measurements in Great-Britain and eight papers (e.g. [Laws and Parsons 1943] and [Marshall and Palmer 1948]), and amount to 1.30, 0.232, 2.25, 67 and 0.846, respectively. The [Best 1950] spectrum is used in [Wessels 1972] to analyse raindrop spectra measured at De Bilt (NL) during 1968 and 1969. The best regression of the 533 observations was found with and . 90% of the observations had a constant ranging from 0.88 to 1.77 (with ). Individual raindrop spectra can thus vary widely in shape, as is illustrated in figure 2.11.

**Summary**
The three presented raindrop spectra can be summarised by the following
formula for the raindrop mass concentration spectrum:

Marshall and Palmer: | |

Ulbrich: | |

Best: | |

Note that in eq. 2.24: [kg m]. |

© 2002 Fabien J.R. van Mook

ISBN 90-6814-569-X

Published as issue 69 in the