The Molecular Diffusion of Ice Crystals of Various Shapes

Two main controlling factors of ice crystal growth are the heat and (gaseous) mass transfers, H&MT, as characterized by the Nusselt number (Nu) for heat and the Sherwood (Sh) number for mass. Nu and Sh express the increase of the molecular diffusions by the relative motions of the particles in air as characterized by the Reynolds number (Re). Traditionally Nu is assumed to be identical to Sh. Past laboratory measurements with various ice crystal shapes by the Toronto Cloud Physics Group involved Sh, thus this study will deal with Sh. The growth of a crystal depends on its size and shape, its fall speed (through Re) and the flux of water vapour from the surrounding air. It is controlled by the diffusion of heat which carries away the energy released on the crystal surface by deposition. Sh is a function of Re and was established by Schemenauer and List (1978)[S&L78] for snow crystals and graupel. Their study, however, did not address the case of pure diffusion (i.e. Re=0). Such values based on approximations have been available in the literature (McDonald, 1963[Mc63], Jayaweera (1971)) only for a very limited number of shapes of crystals. Thus, it was decided to take a general approach to numerically calculate ShRe=0 for any ice crystal with a rectilinear shape. New values of ShRe=0 for additional shapes of crystals are addressed in this study along with the method developed for the computations. The steady state diffusion is controlled by the Laplace equation. It was solved numerically with Dirichlet boundary conditions on a rectilinear 3 dimensional lattice system with variable lattice separation distances for hexagonal plates, hexagonal cylinders, stellar crystals, capped columns, and broadbranched crystals. 2. ELECTROSTATIC ANALOGY