Fluctuations and Luck in Droplet Growth by Coalescence

Alexander B. Kostinski
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Raymond A. Shaw
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After the initial rapid growth by condensation, further growth of a cloud droplet is punctuated by coalescence events. Such a growth process is essentially stochastic. Yet, computational approaches to this problem dominate and transparent quantitative theory remains elusive. The stochastic coalescence problem is revisited and it is shown, via simple back-of-the-envelope results, that regardless of the initial size, the fastest one-in-a-million droplets, required for warm rain initiation, grow about 10 times faster than the average droplet. While approximate, the development presented herein is based on a realistic expression for the rate of coalescence. The results place a lower bound on the relative velocity of neighboring droplets, necessary for warm rain initiation. Such velocity differences may arise from a variety of physical mechanisms. As an example, turbulent shear is considered and it is argued that even in the most pessimistic case of a cloud composed of single-sized droplets, rain can still form in 30 min under realistic conditions. More importantly, this conclusion is reached without having to appeal to giant nuclei or droplet clustering, only occasional “fast eddies.” This is so because, combined with the factor of 10 accelerated growth of the one-in-a-million fastest droplets, the traditional turbulent energy cascade provides sufficient microshear at interdroplet scales to initiate warm rain in cumulus clouds within the observed times of about 30 min. The simple arguments presented here are readily generalized for a variety of time scales, drizzle production, and other coagulation processes.

Department of Physics, Michigan Technological University, Houghton, Michigan

A supplement to this article is available online (DOI: 10.1175/BAMS-86-2-Kostinski)

CORRESPONDING AUTHORS: Alexander B. Kostinski or Raymond A. Shaw, Dept. of Physics, Michigan Tech, 1400 Townsend Dr., Houghton, MI 49931, E-mail: Kostinski: alex_kostinski@mtu.edu; Shaw: rashaw@mtu.edu

After the initial rapid growth by condensation, further growth of a cloud droplet is punctuated by coalescence events. Such a growth process is essentially stochastic. Yet, computational approaches to this problem dominate and transparent quantitative theory remains elusive. The stochastic coalescence problem is revisited and it is shown, via simple back-of-the-envelope results, that regardless of the initial size, the fastest one-in-a-million droplets, required for warm rain initiation, grow about 10 times faster than the average droplet. While approximate, the development presented herein is based on a realistic expression for the rate of coalescence. The results place a lower bound on the relative velocity of neighboring droplets, necessary for warm rain initiation. Such velocity differences may arise from a variety of physical mechanisms. As an example, turbulent shear is considered and it is argued that even in the most pessimistic case of a cloud composed of single-sized droplets, rain can still form in 30 min under realistic conditions. More importantly, this conclusion is reached without having to appeal to giant nuclei or droplet clustering, only occasional “fast eddies.” This is so because, combined with the factor of 10 accelerated growth of the one-in-a-million fastest droplets, the traditional turbulent energy cascade provides sufficient microshear at interdroplet scales to initiate warm rain in cumulus clouds within the observed times of about 30 min. The simple arguments presented here are readily generalized for a variety of time scales, drizzle production, and other coagulation processes.

Department of Physics, Michigan Technological University, Houghton, Michigan

A supplement to this article is available online (DOI: 10.1175/BAMS-86-2-Kostinski)

CORRESPONDING AUTHORS: Alexander B. Kostinski or Raymond A. Shaw, Dept. of Physics, Michigan Tech, 1400 Townsend Dr., Houghton, MI 49931, E-mail: Kostinski: alex_kostinski@mtu.edu; Shaw: rashaw@mtu.edu
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