The Growth of Ice Crystals by Molecular Diffusion

Abstract

The mass transfer of water molecules by diffusion onto ice particles is best described by their Sherwood number (Sh), a dimensionless quantity, which combines molecular and convective effects and depends on the airflow as represented by the Reynolds number (Re). While Sh (Re > 0) has been previously measured in experiments for typical crystal shapes, the limiting case of pure molecular diffusion (Sh0) for zero flow with Re = 0 is not known well and needs independent determination. The direct numerical solution of the controlling Laplace equation links diffusion with electric fields through the electrostatic analogy. It will be solved for the electrostatic potential V around a crystal-shaped conductor of capacitance C. The results will then be converted by similarity theory. This led to the first numerical determination of Sh0 for hexagonal plates, hexagonal columns, stellar crystals, capped columns, and broad-branched crystals. The new data represent another necessary step in the formulation of an experiment-based theory of the growth of freely falling ice crystals in the atmosphere. A discrete version of Gauss's flux law is developed to compute the flux generated by a crystal-shaped conductor in a finite Cartesian grid box, using a Gauss?Seidel iterative scheme. This method is general and can be applied to compute Sh0 for any rectilinear shapes to any degree of accuracy. The dimensionless mass transfer by molecular diffusion, Sh0, is identical to the diffusion of heat characterized by the Nusselt number Nu0.