Solutions to the Droplet Collection Equation for Polynomial Kernels

Abstract

Numerical solutions to the droplet collection equation, using certain polynomial approximations to the gravitational collection kernel, are examined to learn whether they usefully describe the evolution of a cloud droplet size distribution. The results for typical continental and maritime clouds show that the distribution is closely described if the kernel is replaced by 9.44?109(x2+y2),R≤50 ?m; 5.78?103(x+y),R>50 ?m,or by 1.10?1010x2, R≤50 ?m; 6.33?103x,R>50 ?m,where R is the radius of the larger droplet, x its volume in cubic centimeters, and y the volume of the smaller droplet. From the standpoint of including collision and coalescence of droplets in multi-dimensional cloud models an analytic solution to the collection equation is desirable. An attempt should be made to find such solutions based upon either of the above approximations. If these cannot be found because of the piecewise nature of the approximations, then solutions based on the portions for R≤50 ?m would still describe the first few hundred seconds of droplet growth. A comparatively poor description of the droplet distribution comes from the most physically realistic analytic solution presently existing, based on the kernel approximation B(x+y)+Cxy.