Raindrop size distribution

The most well-known study about raindrop size distribution is from Marshall and Palmer done at McGill University in Montréal in 1948.[4] They used stratiform rain with ${\displaystyle \mu =0}$ and concluded to an exponential drop size distribution. This Marshall-Palmer distribution is expressed as:

${\displaystyle N(D)_{MP}=N_{0}e^{-\Lambda D}}$

Where

• N₀ = 8000 m⁻³mm⁻¹ ;
• ${\displaystyle \scriptstyle \Lambda }$ = 4.1 R^-0.21mm⁻¹ (equivalent to 41 R^-0.21cm⁻¹ in the reference[4]), R being the rainrate in stratiform precipitation in millimeters per hour;
• D = raindrop diameter in mm

The units of N₀ are sometimes simplified to cm ⁻⁴ but this removes the information that this value is calculated per cubic meter of air.

As the different precipitations (rain, snow, sleet, etc...), and the different types of clouds that produce them vary in time and space, the coefficients of the drop distribution function will vary with each situation. The Marshall-Palmer relationship is still the most quoted but it must be remembered that it is an average of many stratiform rain events in mid-latitudes.[4] The upper figure shows mean distributions of stratiform and convective rainfall. The linear part of the distributions can be adjusted with particular ${\displaystyle \scriptstyle \Lambda }$ of the Marshall-Palmer distribution. The bottom one is a series of drop diameter distributions at several convective events in Florida with different precipitation rates. We can see that the experimental curves are more complex than the average ones, but the general appearance is the same.

Many other forms of distribution functions are therefore found in the meteorological literature to more precisely adjust the particle size to particular events. Over time researchers have realized that the distribution of drops is more of a problem of probability of producing drops of different diameters depending on the type of precipitation than a deterministic relationship. So there is a continuum of families of curves for stratiform rain, and another for convective rain.[4]

The Marshall and Palmer distribution uses an exponential function that does not simulate properly drops of very small diameters (the curve in the top figure). Several experiments have shown that the actual number of these droplets is less than the theoretical curve. Carlton W. Ulbrich developed a more general formula in 1983 taking into account that a drop is spherical if D <1 mm and an ellipsoid whose horizontal axis gets flattened as D gets larger. It is mechanically impossible to exceed D = 10 mm as the drop breaks at large diameters. From the general distribution, the diameter spectrum changes, μ = 0 inside the cloud, where the evaporation of small drops is negligible due to saturation conditions and μ = 2 out of the cloud, where the small drops evaporate because they are in drier air. With the same notation as before, we have for the drizzle the distribution of Ulbrich:[3]

${\displaystyle N_{0}\mathrm {(cm^{-4-\mu })} =\left[{\frac {6}{\pi (\mu +3)!}}\right]\left({\frac {M_{l}}{10^{-3}\rho _{e}}}\right)\Lambda ^{\mu +4}}$ and ${\displaystyle \Lambda \mathrm {(cm^{-1})} ={\frac {3.67+\mu }{D_{0}}}}$

Where ${\displaystyle M_{l}}$ is the liquid water content, ${\displaystyle \rho _{e}}$ water density, and ${\displaystyle \scriptstyle D_{0}\approx }$ 0.2 is an average value of the diameter in drizzle. For rain, introducing rainrate R (mm/h), the amount of rain per hour over a standard surface:[3]

${\displaystyle D_{0}{(cm)}\approx 0.13R^{0.14}}$ and ${\displaystyle N_{0}\mathrm {(cm^{-4-\mu })} \approx 6\times 10^{-2}\exp(3.2\times \mu )}$