An Analytic Scaling Law for the Depositional Growth of Snow in Thin Mixed-Phase Layer Clouds

Abstract

In various practical problems, such as assessing the threat of aircraft icing or calculating radiative transfer, it is important to know whether mixed-phase clouds contain significant liquid water content. Some mixed-phase clouds remain predominantly liquid for an extended time, whereas others glaciate quickly. The glaciation rate of mixed-phase layer clouds is thought to depend on various factors, including number concentration of snow crystals, terminal velocity of snow crystals, and crystal habit type. This paper attempts to quantify some of these factors by deriving scaling laws (i.e., power laws) for the mixing ratio and sedimentation flux of snow at cloud base. The scaling laws are derived from the governing equation for snow mixing ratio. They neglect aggregation of snow crystals and accretion of supercooled liquid by snow crystals. The scaling laws permit arbitrary exponents and prefactors of the mass?diameter and fall speed?diameter power laws, allowing flexibility in crystal habit properties. The scaling laws are tested using idealized large-eddy simulation (LES) of three thin, midlevel layer clouds. The scaling laws agree adequately with the LES over one order of magnitude for snow flux and over two orders of magnitude for snow mixing ratio. They indicate, for instance, that in the present LES, cloud-base snow flux and snow mixing ratio increase faster than linearly with increasing cloud thickness and supersaturation with respect to ice. By varying the exponents and prefactors of the scaling laws, one may explore the sensitivity of glaciation rate to habit type. The relationship is complex, but, for the cloud cases examined, dendrites tend to glaciate cloud more rapidly than plates.