Applications of a Fast Computational Method for Telford's Theory of Cloud Droplet Coalescence Growth

Abstract

A fast computational method for evaluating Telford's simplified stochastic model of the coalescence process of cloud droplet growth is shown to be useful for approximating the spectrum development in the early stages of precipitation, particularly for cases of narrow droplet size spectra. The method uses a saddle-point integration of the Laplace transform of the growth probability function, and turns out to be at least two orders of magnitude faster than the usual kinetic equation computations. Simple, rough approximations for large and small time intervals are developed. Instead of setting the collection efficiency E = 1 as in Telford (1955), realistic formulas for E taken from the Scott-Chen (1970) formula, the Neiburger-Lee-Lobl-Rodriguez (Lee, 1975) formula, or a Lagrange interpolation to fit the Davis-Sartor and Schafrir-Neiburger theories are used, yielding considerably different conclusions from those of Telford. Davies' and Beard and Pruppacher's terminal velocity formulas were both used, with little difference found between them. A simple modification of our method was developed to treat the continuous collection model for a single particle failing in a uniform cloud of droplets. The results show that the continuous and discrete models have nearly the same growth rate when the growing droplets are larger than 50 µm in radius; the probability curves of the two models have nearly the same behavior. Consequently, the growth rate calculated by the continuous model is used for droplets from 50 to 500 µm. A typical result shows that 0.01% of the droplets with an initial size of 30 µm can grow to a size of 400 µm within 30.77 min. Three examples which consider a single large drop size and either a single or narrow discrete spectrum of smaller droplet sizes are presented for the comparison of Telford's model with a kinetic equation calculation, arranged to treat volume categories which are integral multiples of a smallest size. An estimate of the range of validity of Telford's model as an approximation to the calculation of the kinetic equation is discussed. Under the same initial conditions, the calculation of the kinetic equation in FORTRAN needs about 50 s of computer time but the modified Telford method needs only 0.2 s in BASIC.