Illustration of precipitation distributions as an open system at steady state, where the time scales for condensation, collection, and precipitation are equal, and net transfer of mass into precipitation size bins is balanced by fallout.

Derived rain size distributions grouped by three values of liquid water path L, updraft velocity w, and gamma function prefactor exponent μ. Black dashed lines represent the Marshall–Palmer distribution for rain rates starting at 0.01 mm/h and increasing by orders of magnitude to 100 mm/h.

Illustration of precipitation distributions as an open system at steady state, where the time scales for condensation, collection, and precipitation are equal, and net transfer of mass into precipitation size bins is balanced by fallout.

Derived rain size distributions grouped by three values of liquid water path L, updraft velocity w, and gamma function prefactor exponent μ. Black dashed lines represent the Marshall–Palmer distribution for rain rates starting at 0.01 mm/h and increasing by orders of magnitude to 100 mm/h.

Science

A simple explanation that is consistent with observations is provided for what determines how many rain drops fall of varying sizes. A key parameter is the ratio of updraft velocity to how much water is in the cloud.

Impact

Observations were first made of the numbers and sizes of raindrops and snowflakes in the 1940s, germinating the question of why it is that their size distributions take on a particularly simple mathematical form. This study helps to resolve the problem with a straightforward derivation that can be used to link the sizes of precipitation particles -- whether small drizzle or heavy rain -- to the types of clouds that formed them.

Summary

Analytical solutions are derived for the steady-state size distributions of precipitating rain and snow particles assuming growth via collection of suspended cloud particles. Application of the Liouville equation to the transfer of precipitating mass through size bins in a “cascade” yields a characteristic gamma distribution with a Marshall–Palmer exponential tail with respect to particle diameter. For rain, the principle parameters controlling size distribution shape are cloud droplet liquid water path and cloud updraft speed. Stronger updrafts lead to greater concentrations of large precipitating drops and a peak in the size distribution. The solutions provide a means for relating size distributions measured in the air or on the ground to cloud bulk microphysical and dynamic properties.