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On the Evaluation of the Collection Kernel for the Coalescence of Water Droplets

Alexis B. LongDivision of Cloud Physics, CSIRO, Sydney, Australia

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Michael J. MantonDept. of Mathematics, Monash University, Melbourne, Australia

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Abstract

Two anomalies are described which arise in the kernel for stochastic droplet collection when it is specified by the formula of Scott and Chen for the linear collision efficiency y(R,r) and by the formula of Wobus et al. for the droplet terminal velocity V(R). It is pointed out that if accurate values for y(R,r) are to be obtained for a given droplet pair by interpolation using data for specific droplet pairs, then for large droplet radii (R<30 μm) it is desirable that these data he tabulated for 2-μm intervals of R. It is shown that if the difference in terminal velocities of two droplets is computed from a formula approximating V(R) and composed of various functions V*(R) applicable over adjoining domains of R, then it is necessary that these functions be constructed so that the formula and its derivatives, at least up to second order, are everywhere continuous. An improved formula for V(R) satisfying this criterion is described.

Abstract

Two anomalies are described which arise in the kernel for stochastic droplet collection when it is specified by the formula of Scott and Chen for the linear collision efficiency y(R,r) and by the formula of Wobus et al. for the droplet terminal velocity V(R). It is pointed out that if accurate values for y(R,r) are to be obtained for a given droplet pair by interpolation using data for specific droplet pairs, then for large droplet radii (R<30 μm) it is desirable that these data he tabulated for 2-μm intervals of R. It is shown that if the difference in terminal velocities of two droplets is computed from a formula approximating V(R) and composed of various functions V*(R) applicable over adjoining domains of R, then it is necessary that these functions be constructed so that the formula and its derivatives, at least up to second order, are everywhere continuous. An improved formula for V(R) satisfying this criterion is described.

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