Ice multiplication by fragmentation during collision-freezing of supercooled rain or drizzle is investigated. A zero-dimensional dynamical system describes the time evolution of number densities of supercooled drops and ice crystals in a mixed-phase cloud. The characteristic time scale for this collision-freezing ice fragmentation is controlled by the collision efficiency, the number of ice fragments per freezing event, and the available number concentration of supercooled drops. The rate of the process is proportional to the number of supercooled drops available. Thus, ice may multiply extensively, even when the fragmentation number per freezing event is relatively small. The ratio of total numbers of ice particles to those from the first ice, namely, the "ice-enhancement factor,"is controlled both by the number of fragments per freezing event and by the available number concentration of supercooled drops in a similar manner. Especially, when ice fragmentation by freezing of supercooled drops is considered in isolation, the number of originally existing supercooled drops multiplied by the fragmentation number per freezing event yields the eventual number of ice crystals. When supercooled drops are continuously generated by coalescence, ice crystals from freezing fragmentation also continuously increase asymptotically at a rate equal to the generation rate of supercooled drops multiplied by the fragmentation number per freezing event. All these results are expressed by simple analytical forms, thanks to the simplicity of the theoretical model. These parameters can practically be used as a means for characterizing observed mixed-phase clouds.
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