A Simple Perturbation Model for the Electrostatic Shape of Falling Drops

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Abstract
A perturbation model for the shape of falling drops in the presence of electric fields and charges was developed by extension of previous methods that includes aerodynamic effects in the pressure balance equation of Laplace. Use of a consistent first-order perturbation equation and spherical harmonies helped to reconcile apparent inconsistencies in the cosine series method originated by Savic and modified by Pruppacher and Pitter. The electric stress was included in the first-order model using a sphere in a uniform vertical field, with the effect of a net charge approximated by a reduction in surface tension for a spherical drop. The perturbation drop shapes for electric distortion were found to be appreciably altered by coupling between shape and fall speed. A simple axis ratio formula, valid for small perturbation was based on the principal shape coefficient, containing term for the aerodynamic distortion, electric field and charge: α = 1 − (0.1 98 We-0.281X2)FQ. This equation, when coupled with a shape-drag relation, yielded a cubic equation for axis ratio. Calculated axis ratios were consistent with wind tunnel observations for non-oscillating drops in vertical electric fields and generally agreed with the numerical findings of Chuang for electrostatic-aerodynamic shapes. Our results indicate detectable shape changes in thunderstorms—produced directly by strong electric fields, but modified considerably by changes in fall speed, with and without charges. The most favorable situation appears to be in vertical fields just before initiation of lightning where raindrops could be nearly spherical producing ZDR changes as large as several dB.

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