Explosive ice multiplication by mechanical break-up in ice-ice collisions: A dynamical system-based study

Mechanical break-up in ice-ice collisions observed in the laboratory can lead to explosive ice multiplication. The possibility is examined in detail by constructing an idealized analytical single-point (zero-dimension) model assuming spatial homogeneity within a cloud. The rate of generation of primary ice is fixed at a constant value over time, corresponding to a situation in which the relative humidity (at water saturation as for a mixed-phase cloud) and updraught speed are fixed. This would further imply an infinite supply of vapour, if the crystal concentration were somehow infinite. Fixed times are assumed for the transformation of the generated primary ice into small graupel, which then grow into large graupel. Secondary ice particles produced by collisions between small and large graupel, in turn, potentially multiply explosively, because the secondary ice may eventually grow to become splintering graupel with a positive feedback. Two basic processes can lead to explosive ice multiplication. The first process is the generation of primary ice, which initiates ice multiplication only with a relatively low threshold (∼10^-4 m^-3 s^-1) compared to typical observed values for deep convective clouds. Then the second process is induced by a sufficiently large initial number of ice crystals, which generate enough both small and large graupel particles, leading to explosive multiplication, even when the rate of generation of primary ice is below the threshold. These initial crystals at super-critical amounts may be from any source. Only a low number density of crystals of the order of 1 m^-3 (or 10^-3 l^-1) is required initially. In both cases, the ice number literally explodes within a finite time, with a time-scale from 30 to 200 min under the idealized single-point model studied. Importantly, in real clouds a typical number density of large graupel is above the threshold needed for explosive break-up.