I was walking on the terrace the other day when it started drizzling. As I was making haste, I noticed something nobody would pay attention to. The wet spots left by the raindrops on the floor came in all sizes. And I wondered what is the size distribution of the raindrops? For example, something like 60% of the raindrops are smaller than 3 millimeters, 30% of the raindrops are bigger than 3 millimeters but smaller than 1 centimeter, and 10% are greater than a centimeter. I wondered if there was some mathematical law.
To answer this question, we must somehow measure the size of the raindrops falling from the sky. This seemed hopeless. A better way would be to measure the wet spots left by the rain, but the small ones dry up as soon as they appear.
To my relief, a German meteorologist J. Wiesner, in 1895, developed a clever method to permanently capture the spots left by rain [1]. An absorbent paper is coated in dye, and allowed to dry. When exposed to rain for a few seconds and dried, the raindrops leave permanent spots on the paper. A big raindrop would leave a big spot, so measuring the size of the spot on paper gives an estimate of the actual size of the raindrop. Tabulating the number of spots and the radius of each spot gives us the size distribution.
For example, consider a hypothetical absorbent paper as shown below.
There are 21 spots in the above figure. It is most convenient to represent their sizes in a table.
| Radius of Spot | Number of Spots |
| 5 mm | 10 |
| 7 mm | 2 |
| 8 mm | 2 |
| 10 mm | 2 |
| 11 mm | 1 |
| 15 mm | 3 |
| 20 mm | 1 |
If we say spots with a radius greater than or equal to 15 mm are large, then about 19% of the drops are large. If spots in the range of 10 mm to 15 mm are categorized as medium, then about 14.3% of the drops are medium. If spots smaller than 10 mm are categorized as small, then about 66.7% of the drops are small. This is the size distribution.
In America, Wilson A. Bentley was oblivious to what his European contemporaries were doing. He developed an ingenious method to measure the size of raindrops [2]. Imagine dropping a drop of water in a bowl of flour. After a while, the drop produces a flour pellet. This is the principle behind Bentley’s method. The procedure used is described in a paper by Laws and Parsons [3]:
“To collect a sample, the observer sifted calibrated flour into the pans and scraped off the excess flour so as to leave a smooth, uncompacted surface. Prepared pans were not allowed to stand for more than two hours without resifting. The observer placed the pan on the palm of one hand and shielding the flour with a large cover held in the other hand, stepped out on the flat roof of the laboratory. The cover was then snatched away while the pan was held well away from his body so that the sample would not be contaminated with splashes from the observer’s hat. When the flour was moderately sprinkled, the cover was replaced. A record was made of the duration of exposure and the hour and minute of the day.”
J. O. Laws and D. A. Parsons, “The relation of raindrop size to intensity,” Proceedings of the 24th Annual Meeting Transactions of American Geophysical Union, 1943.
The raindrops collected on the flour pan are allowed to dry and they form pellets. A big raindrop results in a big pellet, a small raindrop a small pellet. Measuring the sizes of the various pellets, we get a rough idea of the size distribution of raindrops.
Laws and Parsons collected data in tables similar to ours, but there was no mathematical equation for the size distribution in their work. J. S. Marshall and W. McK. Palmer, from McGill University, collected Laws and Parsons’s data and discovered the approximate mathematical law [4].
According to this law, larger drops are rare. The larger the drop, the lesser its frequency. To be more precise, the distribution is a decaying exponential. This answered my question but raised another. Is there a reason for the exponential decay? Can this mathematical law be derived from something basic?
This was answered in a 2009 Nature Physics paper [4]. The authors, Emmanuel Villermaux and Benjamin Bossa, tried to recreate a raindrop falling from the sky in a laboratory. A drop of water was dropped into a jet of air traveling upward. The fall of the drop is recorded on video (which you can find in the Supplementary Information of [4]). The authors built a mathematical model based on the motion of the water drop through the jet of air. Using this model, they were able to explain the exponential decay. I was thrilled by this and wrote this post.
References
[1] J. Wiesner, “Beiträge zur Kenntnis des tropischen Regens, (Contributions to the knowledge of the tropical rain),” Sitzungsberichte. Kaiserliche Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Klasse, vol. 104, pp. 1397–1434, 1895.
[2] W. A. Bentley, “Studies of raindrops and raindrop phenomena,” Monthly Weather Review, vol. 32, no. 10, pp. 450–456, 1904.
[3] J. O. Laws and D. A. Parsons, “The relation of raindrop size to intensity,” Proceedings of the 24th Annual Meeting Transactions of American Geophysical Union, pp. 452–460, 1943.
[4] Villermaux, E., Bossa, B. Single-drop fragmentation determines size distribution of raindrops. Nature Phys 5, 697–702 (2009). https://doi.org/10.1038/nphys1340
Rookie Physics 