# Size of a raindrop

Thinking about the fact that raindrops come with a typical size I was wondering how this can be determined. I am pretty sure that the friction with air and the quantity of water in the clouds are involved with the upper limit of the raindrop size but I cannot figure out the details: maybe the friction is the sole responsible, or maybe is the contrary and the clouds are not thick enough to produce raindrops larger than a bit.

Also, what determines the minimum size? Maybe this side of the problem does not involve meteorology and could be explained by surface tension and other static characteristics. Edit: as noted in a comment, this could also be influenced by the dust particle seeding the formation of the raindrop.

For the maximum size I was thinking of computing a length involving air viscosity $$\gamma$$, air and water density $$\rho_{1/2}$$ and gravitational acceleration $$g$$. By dimensional analysis we have:

$$[\gamma]=\frac{[M]}{[T]}\qquad[\rho]=\frac{[M]}{[L]^3}\qquad[g]=\frac{[L]}{[T]^2}$$ And therefore if we want to build a length we will use a product: $$[\gamma^{\alpha}\rho^{\beta}g^{\lambda}]=[M]^{\alpha+\beta}[L]^{-3\beta+\lambda}[T]^{-\alpha-2\lambda}\equiv[L]$$ Which implies $$\alpha=-\beta=-2\lambda$$ and $$\lambda=1+3\beta\;$$, so that $$\;\lambda=-\frac{1}{5}\;,\;\beta=-\alpha=-\frac{2}{5}$$. So the size should be something like: $$d\sim\left(\frac{\gamma^2}{g\rho_{..}^2}\right)^{1/5}\cdot f\left(\frac{\rho_1}{\rho_2}\right)$$ Where $$f$$ is an unknown function and I cannot understand which density (or product of densities) will go in the dimensional factor.

Clearly dimensional analysis cannot do all the work here, moreover I would regard the resulting size as a typical size rather than an upper bound.

So the question remains: how do we determine the upper (and the lower) bound on the size of a raindrop?

Thanks for any help!

Edit: in the dimensional computation we should at least consider also the surface tension, $$\tau$$. This has dimension of a force: $$[\tau]=\frac{[M][L]}{[T]^2}$$. This will add even more freedom in the guess-result for $$d_{max}$$, thus making some physical reasoning even more needed.

• Aren't raindrops formed by moisture forming on a particle? – user207455 Apr 28 at 8:33
• @SolarMike yeah you are right. I don't think the upper bound should be influenced by the size of the seed particle, but maybe the lower could be.. – AoZora Apr 28 at 8:56
• Given that I have seen really fine rain and big fat drops, do you think wind speed is possibly a factor to include? – user207455 Apr 28 at 8:57
• @SolarMike Yeah in the presence of wind we should have an additional effect resembling an increase in either speed of raindrops or friction. I think that the simplest thing to understand is the wind-less case though. – AoZora Apr 28 at 9:02
• See Prupacher and Klett, Microphysics of Clouds and Precipitation – Chet Miller Apr 28 at 11:26