Terminal velocities of droplets and crystals: Power laws with continuous parameters over the size spectrum

Abstract

This paper presents a unified treatment of cloud particle fall velocities for both liquid and crystalline cloud particles over the entire size range observed in the atmosphere. The fall velocity representation is formulated in terms of the Best (or Davies) number X, and the Reynolds number Re. For the power-law representations used in many applications, the coefficients are found as the continuous analytical functions of X (or diameter) over the entire hydrometeor size range. Analytical asymptotic solutions are obtained for these coefficients for the two regimes that represent large and small particles and correspond to potential and aerodynamical flows, respectively. The new formulation is compared with experimental data and previous formulations for small drops, large nonspherical drops, and various ice crystal habits. For ice crystals, published mass-dimension and area-dimension relationships are used. The advantage of the new representation of fall velocities over previous representations is that the continuous representation avoids inaccuracy at the points of discontinuity for different size regimes, allows easier parameterization of the hydrometeor size spectra, and allows for continuous integration over the size spectrum. The new fall velocity formulation may be applied to bin-resolving and bulk microphysical models, as well as to remote sensing.