AbstractIce particle multiplication in shallow supercooled clouds is described by a stochastic process, which depends on collisions between ice particles and supercooled water droplets and drops. The multiplication is a consequence of the riming and drop splintering processes, or possibly just one of them. In the riming process, a large ice particle collects water droplets which eject splinters on freezing. In the drop splintering process, a supercooled drop which collides with a splinter, freezes and may eject splinters. Account is taken of the stronger evidence of significant drop splintering for small supercooled drops by considering those drops with diameters less than 250μm as a separate group. The probability of collision between an ice splinter and a water drop is estimated by considering their fall velocities and assuming a collection efficiency of unity. As a consequence of the small and large water drop division, the ice splinters are also divided; small ice splinters collide with all water drops, but large splinters having fall velocities comparable with those of small water drops collide only with large water drops. The growth of all ice particles by diffusion of water vapour or accretion of water droplets is taken into account, splinters not captured by water drops grow to rimer status after an appropriate time. Rimers are removed from the system at the time they fall through the 0°C isotherm. A rimer may also be formed as a result of the freezing of a large supercooled water drop, following its collision with an ice splinter. As the early production of splinters by such a rimer depends on its initial mass, which equals that of the drop, a distribution of large water drop masses is considered.Generating functions are formulated for the probabilities of particular numbers of the various categories of ice particle existing at a general timet, as a consequence of one initial ice particle. These functions lead to a set of renewal equations for the estimated mean numbers of ice particles at timet. The Laplace transforms of the renewal equations are studied and it is shown that a simple large time analysis provides a valid description of the growth mechanism during most of the cloud lifetime. A small time behaviour marching technique completes the growth description. The large time analysis predicts thatA1exp(p0t) ice particles exist at timetas a consequence of one initial small ice splinter. Values of the growth parameter,p0, are given for many combinations of the physical parameters. An integration of this large time result provides a description of ice particle multiplication initiated by the nucleating particles introduced into a cloud by a constant updraught. A thermal model is also considered and results in growth times near to those obtained for the constant updraught model. These growth times are considerably less than the growth times predicted by earlier workers.The field observations of several workers are all found to be consistent with this analysis. In many situations the most likely explanation of the multiplication involves two processes. These are: splinter production by riming; and the capture of ice splinters by drops, the drops becoming riming agents when they freez